How Leverage Shifts and Scales a Volatility Skew
Author | : Roger Lee |
Publisher | : |
Total Pages | : 15 |
Release | : 2015 |
Genre | : |
ISBN | : |
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To model leveraged investments such as leveraged ETFs, define the beta-leveraged product on a positive semimartingale S to be the stochastic exponential of beta times the stochastic logarithm of S.In various asymptotic regimes, we relate rigorously the implied volatility surfaces of the beta-leveraged product and the underlying S, via explicit shifting/scaling transformations. In particular, a family of regimes with jump risk admit a shift coefficient of -3/2, unlike the previously conjectured 1/2 shift. The 1/2, we prove, holds in a family of continuous (including fBm-driven) stochastic volatility regimes at short expiry and at small volatility-of-volatility. In another regime, which does not admit a simple spatial shifting/scaling rule, we find an expiry scaling together with a spatial transformation.